Combinations of Variables

Another technique that is sometimes employed to reduce partial differential equations to ordinary differential equations is combination of variables or a similarity transformation. 

The process of normalization can be used to establish the applicability of combining the independent variables of the given PDE. For example, consider the flow of a fluid near a wall suddenly set in motion. Following [1], the problem statement is as follows  

A semi-infinite body of liquid with constant density (p) and viscosity (jx) is bounded on one side by a flat surface (the xz-plane). Initially, the fluid and 

One would like to know the velocity profile as a function of y and t. If there is no pressure gradient or gravity force in the x-direction and the flow is laminar, then this simplified problem can be solved in the following way. 

Note that the initial and boundary conditions are combined to form Equation 6.105. Equation 6.103 solves to give the general solution 

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For a reaction as complex as catalytic enantioselective cyclopropanation with zinc carbenoids, there are many experimental variables that influence the rate, yield and selectivity of the process. From an empirical point of view, it is important to identify the optimal combination of variables that affords the best results. From a mechanistic point of view, a great deal of valuable information can be gleaned from the response of a complex reaction system to changes in, inter alia, stoichiometry, addition order, solvent, temperature etc. Each of these features provides some insight into how the reagents and substrates interact with the catalyst or even what is the true nature of the catalytic species. 

Using Table 52 the variables are El(FL ), L(L), d(L), (d - d,)(L), T(FL), and P(F). Note that this I is moment-area which is in the units of ft (not to be confused with I given in Table 52 which is moment of inertia, see Chapter 2, Strength of Materials, for clarification). The number of FI ratios that will describe the problem is equal to the number of variables (6) minus the number of fundamental dimensions (F and L, or 2). Thus, there will be four FI ratios (i.e., 6-2 = 4), FI, flj, fl, and FI. The selection of the combination of variables to be included in each n ratio must be carefully done in order not to create a complicated system of ratios. This is done by recognizing which variables will have the fundamental dimensions needed to cancel with the fundamental dimensions in the other included variables to have a truly dimensionless ratio. With this in mind, FI, is... 

Simulation in general describes calculations with models, where different options and combinations of variables can be quickly played through. Molecular simulations allow the characterization of molecular properties during the motions of the molecular models, over time. 

Compare the amounts of catalyst needed for 50% conversion with these combinations of variables,... 

This combination of variables occurs so often in physical chemistry, that we give it a name we call it the enthalpy, and give it the symbol H. Accordingly, we rewrite Equation (3.15) as ... 

The method of combination of variables requires that a suitable combination ofy and t can be found. Dimensionally, equation 10.19 can be written as... 

A general alternative to stepwise-type searching methods for variable selection would be methods that attempt to explore as much of the possible solution space as possible. An exhaustive search of all possible combinations of variables is possible only for problems that involve relatively few x variables. However, it... 

In addition to the graphical representations we also obtain a set of simple linear combinations of variables that enable us to... 

Prediction of BOD value. In the ten clusters Identified by the K-means clustering procedure, two clusters were represented by chemicals with only low BOD values and one cluster with nearly all (18 of 19 or 95 %) high BOD values (Table III). Therefore, no discrimination was attempted within these clusters. In the remaining clusters there were 202 high BOD chemicals and 97 low BOD chemicals. Of these, approximately 75 % (152 of 202) were correctly classified Into the high BOD class, while 73 Z (71 of 97) were correctly classified Into the low BOD class. Using both the clustering and discrimination analyses, 77 % (170 of 220) and 78 % (93 of 120) of the chemicals In the data base were correctly classified. Within each of the clusters, between 2 and 4 molecular connectivity Indices were used In the final discriminant functions to separate the two classes of BOD. Within each cluster a different combination of variables were used as discriminators. Because of Che exploratory nature of this analysis, we lowered the F-ratlo Inclusion level Co 1.0. In several of the clusters, the F-ratlos for variables Included In Che discriminant functions were subsequently small(e.g., < 4.0). 

Members of the Chapter 3 work group particularly emphasized avoidance of implausible combinations of variables. Some conjectured that substantial benefit may be had just by excluding combinations of variables that are unreasonable, i.e., dependencies in the tails may be substantially more important for practical purposes than dependencies close to the center of a distribution. 

Differences between PIS and PCR Principal component regression and partial least squares use different approaches for choosing the linear combinations of variables for the columns of U. Specifically, PCR only uses the R matrix to determine the linear combinations of variables. The concentrations are used when the regression coefficients are estimated (see Equation 5.32), but not to estimate A potential disadvantage with this approach is that variation in R that is not correlated with the concentrations of interest is used to construct U. Sometiraes the variance that is related to the concentrations is a verv... 

Various approaches can be taken for constructing the U matrix. With PCR, a principal components analysis is used because PCA is an efficient method for finding linear combinations of variables that describe variation in the row space of R (See Section 4.2.2). With analytical chemistry data, it is usually possible to describe the variation in R using significantly fewer PCs than the number of original variables. This small number of columns effectively eliminates the matrix inversion problem. 

In the field of chemometrics, PCR and PLS are the most widely used of the inverse calibration methods. Tliese methods solve the matrix inversion problem inherent to the inverse methods by using a linear combination of variables in... 

Just as process translation or scaling-up is facilitated by defining similarity in terms of dimensionless ratios of measurements, forces, or velocities, the technique of dimensional analysis per se permits the definition of appropriate composite dimensionless numbers whose numeric values are process-specific. Dimensionless quantities can be pure numbers, ratios, or multiplicative combinations of variables with no net units. 

It appears that the formal theories are not sufficiently sensitive to structure to be of much help in dealing with linear viscoelastic response Williams analysis is the most complete theory available, and yet even here a dimensional analysis is required to find a form for the pair correlation function. Moreover, molecular weight dependence in the resulting viscosity expression [Eq. (6.11)] is much too weak to represent behavior even at moderate concentrations. Williams suggests that the combination of variables in Eq. (6.11) may furnish theoretical support correlations of the form tj0 = f c rjj) at moderate concentrations (cf. Section 5). However the weakness of the predicted dependence compared to experiment and the somewhat arbitrary nature of the dimensional analysis makes the suggestion rather questionable. 

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